I'm confused on what this proof would look like.
I know that by Martingale Convergence Theorem:
$$X= \lim Z_n / \mu^n$$ exists a.s.
I also know that Kesten Stigum Theorem:
Assume that $E[L] >1$ and that $p_1 \neq 1$. Then $X>0$ a.s. on the event of survival iff $E[L\log_{+} L] < \infty$
Is that theorem useful in proving what i want to prove?
Edit: Let $\xi{i}^n , i, n, \geq 1$ be i.i.d. nonnegative integer random variables. Define a sequence $Z_n, n \geq 0 $ by $Z_0 =1$ and
$Z_{n+1} = \xi_1^{n+1}+...+ \xi_{Z_n}^{n+1}$ if $Z_n >0$, $0$ if $Z_n=0$
$Z_n$ is the galton-watosn process.
$ \mu = E\xi_m^i \in (0,\infty)$